Spectral estimates on the sphere

TL;DR

This paper derives optimal spectral estimates for Schrödinger operators on the sphere, linking eigenvalues to potential norms, and explores differences from Euclidean space estimates.

Contribution

It establishes the first eigenvalue bounds on the sphere based on Lebesgue norms of the potential, extending spectral theory to curved geometries.

Findings

Optimal eigenvalue estimates depend on potential norms.

Characterization of semi-classical asymptotic regimes.

Differences between spherical and Euclidean spectral estimates.

Abstract

In this article we establish optimal estimates for the first eigenvalue of Schr\"odinger operators on the d-dimensional unit sphere. These estimates depend on Lebsgue's norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semi-classical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.